Revision: August 21, E Main Suite D Pullman, WA (509) Voice and Fax

Transcription

1 2.7.1: Sinusidal signals, cmplx xpnntials, and phasrs Rvisin: ugust 21, E Main Suit D ullman, W ( ic and Fax Ovrviw In this mdul, w will rviw prprtis f sinusidal functins and cmplx xpnntials. W will als intrduc phasr ntatin, which will significantly simplify th sinusidal stady-stat analysis f systms, and prvid trminlgy which will b usd in subsqunt sinusidal stady-stat rlatd mduls. Mst f th matrial prsntd hr has bn prvidd prviusly in chaptr 2.5.3; this matrial will, hwvr, b imprtant nugh t bar rptitin. Likwis, a brif vrviw f cmplx arithmtic, which will b ssntial in using cmplx xpnntials ffctivly, is prvidd at th nd f this mdul. Studnts wh nd t rviw cmplx arithmtic may find it usful t prus this vrviw bfr rading th sctin f this mdul rlating t cmplx xpnntials and phasrs. fr bginning this mdul, yu shuld b abl t: Writ th quatin gvrning an arbitrary csin functin Sktch th sinusid crrspnding t a givn csin functin rfrm cmplx arithmtic ftr cmplting this mdul, yu shuld b abl t: Dfin pridic signals Dfin th amplitud, frquncy, radian frquncy, and phas f a sinusidal signal Exprss sinusidal signals in phasr frm This mdul rquirs: N/ Sinusidal Signals: Th sinusidal signal shwn in Figur 1 is rprsntd mathmatically by: f ( t cs( ωt (1 Th amplitud r pak valu f th functin is. is th maximum valu achivd by th functin; th functin itslf is bundd by + and -, s that f ( t. Th radian frquncy r angular frquncy f th functin is ω; th units f ω ar radians/scnd. Th functin is said t b pridic; pridic functins rpat thmslvs at rgular intrvals, s that f ( t + nt f ( t (2 whr n is any intgr and T is th prid f th signal. Th sinusidal wavfrm shwn in Figur 1 gs thrugh n cmplt cycl r prid in T scnds. Sinc th sinusid f quatin (1 rpats itslf vry 2π radians, th prid is rlatd t th radian frquncy f th sinusid by: Dc: XXX-YYY pag 1 f 9 Cpyright Digilnt, Inc. ll rights rsrvd. Othr prduct and cmpany nams mntind may b tradmarks f thir rspctiv wnrs.

2 2.7.1: Sinusidal signals, cnmplx xpnntials, and phasrs ω 2π (3 T It is cmmn t dfin th frquncy f th sinusid in trms f th numbr f cycls f th wavfrm which ccur in n scnd. In ths trms, th frquncy f f th functin is: f 1 (4 T Th units f f ar cycls/scnd r hrtz (abbrviatd Hz. Th frquncy and radian frquncy ar rlatd by r quivalntly, ω f (5 2π ω 2πf (6 Rgardlss f whthr th sinusid s rat f scillatin is xprssd as frquncy r radian frquncy, it is imprtant t raliz that th argumnt f th sinusid in quatin (1 must b xprssd in radians. Thus, quatin (1 can b xprssd in trms f frquncy in Hz as: f ( t cs( 2πft (7 T avid cnfusin in ur mathmatics, w will almst invariably writ sinusidal functins in trms f radian frquncy as shwn in quatin (1, althugh Hz is gnrally takn as th standard unit fr frquncy (xprimntal apparatus, fr xampl, cmmnly xprss frquncy in Hz. Figur 1. Csin wavfrm. pag 2 f 9 Cpyright Digilnt, Inc. ll rights rsrvd. Othr prduct and cmpany nams mntind may b tradmarks f thir rspctiv wnrs.

3 2.7.1: Sinusidal signals, cnmplx xpnntials, and phasrs mr gnral xprssin f a sinusidal signal is v( t cs( ω t + θ (8 whr θ is th phas angl r phas f th sinusid. Th phas angl simply translats th sinusid alng th tim axis, as shwn in Figur 2. psitiv phas angl shifts th signal lft in tim, whil a ngativ phas angl shifts th signal right this is cnsistnt with ur discussin f stp functins in chaptr 2.1, whr it was ntd that subtracting a valu frm th unit stp argumnt rsultd a tim dlay f th functin. Thus, as shwn in Figur 2, a psitiv phas angl causs th sinusid t b shiftd lft by ω θ scnds. Th units f phas angl shuld b radians, t b cnsistnt with th units f ω t in th argumnt f th csin. It is typical, hwvr, t xprss phas angl in dgrs, with 180 crrspnding t π radians. Thus, th cnvrsin btwn radians and dgrs can b xprssd as: 180 Numbr f dgrs Numbr f radians π Fr xampl, w will cnsidr th tw xprssins blw t b quivalnt, thugh th xprssin n th right-hand sid f th qual sign cntains a mathmatical incnsistncy: π cs( ω t + cs( ωt θ ω Figur 2. Csin wavfrm with nn-zr phas angl. pag 3 f 9 Cpyright Digilnt, Inc. ll rights rsrvd. Othr prduct and cmpany nams mntind may b tradmarks f thir rspctiv wnrs.

4 2.7.1: Sinusidal signals, cnmplx xpnntials, and phasrs Fr cnvninc, w intrduc th trms lading and lagging whn rfrring t th sign n th phas angl, θ. sinusidal signal v 1 (t is said t lad anthr sinusid v 2 (t f th sam frquncy if th phas diffrnc btwn th tw is such that v 1 (t is shiftd lft in tim rlativ t v 2 (t. Likwis, v 1 (t is said t lag anthr sinusid v 2 (t f th sam frquncy if th phas diffrnc btwn th tw is such that v 1 (t is shiftd right in tim rlativ t v 2 (t. This trminlgy is dscribd graphically in Figur 3. Figur 3. Lading and lagging sinusids. Finally, w nt that th rprsntatin f sinusidal signals as a phas shiftd csin functin, as prvidd by quatin (8, is cmpltly gnral. If w ar givn a sinusidal functin in trms f a sin functin, it can b radily cnvrtd t th frm f quatin (8 by subtracting a phas f 2 π (r 90 frm th argumnt, sinc: π sin( ωt cs( ωt 2 Likwis, sign changs can b accuntd fr by a π radian phas shift, sinc: cs( ω t cs( ωt π Obviusly, w culd hav chsn ithr a csin r sin rprsntatin f a sinusidal signal. W prfr th csin rprsntatin, sinc a csin is th ral part f a cmplx xpnntial. In th nxt mdul, w will s that sinusidal stady-stat circuit analysis is simplifid significantly by using cmplx xpnntials t rprsnt th sinusidal functins. Th csin is th ral part f a cmplx xpnntial (as w saw prviusly in chaptr Sinc all masurabl signals ar ral valud, w tak th ral part f ur cmplx xpnntial-basd rsult as ur physical rspns; this rsults in a slutin f th frm f quatin (8. Sinc rprsntatin f sinusidal wavfrms as cmplx xpnntials will bcm imprtant t us in circuit analysis, w dvt th fllwing subsctin t a rviw f cmplx xpnntials and thir intrprtatin as sinusidal signals. pag 4 f 9 Cpyright Digilnt, Inc. ll rights rsrvd. Othr prduct and cmpany nams mntind may b tradmarks f thir rspctiv wnrs.

5 2.7.1: Sinusidal signals, cnmplx xpnntials, and phasrs Cmplx Expnntials and hasrs: Eulr s idntity can b usd t rprsnt cmplx numbrs as cmplx xpnntials: θ j csθ j sinθ (9 If w gnraliz quatin (9 t tim-varying signals f arbitrary magnitud w can writ: j( ωt+ θ cs( ωt + θ j sin( ωt + θ (10 s that and { j( ω t + θ } cs( ωt + θ R (11 { j( ω t + θ } sin( ωt + θ Im (12 whr { j( ω R t +θ } and { j( ω Im t +θ } f j( ω t+ θ, rspctivly. dnt th ral part f Th cmplx xpnntial f quatin (10 can als b writtn as: j( ω t+θ and th imaginary part j( ωt + θ jωt (13 Th trm n th right-hand sid f quatin (13 is simply a cmplx numbr which prvids th magnitud and phas infrmatin f th cmplx xpnntial f quatin (10. Frm quatin (11, this magnitud and phas can b usd t xprss th magnitud and phas angl f a sinusidal signal f th frm givn in quatin (8. Th cmplx numbr in plar crdinats which prvids th magnitud and phas angl f a timvarying cmplx xpnntial, as givn in quatin (13 is calld a phasr. Th phasr rprsnting cs( ω t + θ is dfind as: θ (14 W will us a capital lttr with an undrscr t dnt a phasr. Using bld typfac t rprsnt phasrs is mr cmmn; ur ntatin is simply fr cnsistncy btwn lctur matrial and writtn matrial bldfac typ is difficult t crat n a whitbard during lctur! Nt: Th phasr rprsnting a sinusid ds nt prvid infrmatin abut th frquncy f th sinusid frquncy infrmatin must b kpt track f sparatly. pag 5 f 9 Cpyright Digilnt, Inc. ll rights rsrvd. Othr prduct and cmpany nams mntind may b tradmarks f thir rspctiv wnrs.

6 2.7.1: Sinusidal signals, cnmplx xpnntials, and phasrs Cmplx rithmtic Rviw: Th bulk f th matrial in this sctin is takn frm chaptr It is rpatd hr fr cnvninc. In ur prsntatin f cmplx xpnntials, w first prvid a brif rviw f cmplx numbrs. cmplx numbr cntains bth ral and imaginary parts. Thus, w may writ a cmplx numbr as: whr a + jb (15 j 1 (16 and th undrscr dnts a cmplx numbr. Th cmplx numbr can b rprsntd n rthgnal axs rprsnting th ral and imaginary part f th numbr, as shwn in Figur 4. (In Figur 4, w hav takn th librty f rprsnting as a vctr, althugh it is rally just a numbr. W can als rprsnt th cmplx numbr in plar crdinats, als shwn in Figur 4. Th plar crdinats cnsist f a magnitud and phas angl θ, dfind as: a b (17 1 b θ tan (18 a Ntic that th phas angl is dfind cuntrclckwis frm th psitiv ral axis. Cnvrsly, w can dtrmin th rctangular crdinats frm th plar crdinats frm { } cs( θ a R (19 { } sin( θ b Im (20 whr th ntatin R { } and { } rspctivly. Im dnt th ral part f and th imaginary part f, Th plar crdinats f a cmplx numbr ar ftn rprsntd in th frm: θ (21 pag 6 f 9 Cpyright Digilnt, Inc. ll rights rsrvd. Othr prduct and cmpany nams mntind may b tradmarks f thir rspctiv wnrs.

7 2.7.1: Sinusidal signals, cnmplx xpnntials, and phasrs sin( θ cs( θ Figur 4. Rprsntatin f a cmplx numbr in rctangular and plar crdinats. n altrnat mthd f rprsnting cmplx numbrs in plar crdinats mplys cmplx xpnntial ntatin. Withut prf, w claim that 1 θ (22 Thus, is a cmplx numbr with magnitud 1 and phas angl θ. Frm Figur 4, it is asy t s that this dfinitin f th cmplx xpnntial agrs with Eulr s quatin: θ j csθ j sinθ (23 With th dfinitin f quatin (22, w can dfin any arbitrary cmplx numbr in trms f cmplx numbrs. Fr xampl, ur prvius cmplx numbr can b rprsntd as: (24 W can gnraliz ur dfinitin f th cmplx xpnntial t tim-varying signals. If w dfin a j t tim varying signal ω, w can us quatin (23 t writ: jωt csωt j sinωt (25 j t ω Th signal can b visualizd as a unit vctr rtating arund th rigin in th cmplx plan; th tip f th vctr scribs a unit circl with its cntr at th rigin f th cmplx plan. This is illustratd in Figur 3. Th vctr rtats at a rat dfind by th quantity ω -- th vctr maks n 2π cmplt rvlutin vry scnds. Th prjctin f this rtating vctr n th ral axis tracs ω ut th signal csω t, as shwn in Figur 3, whil th prjctin f th rtating vctr n th imaginary axis tracs ut th signal sinω t, als shwn in Figur 5. Thus, w intrprt th cmplx xpnntial functin as an altrnat typ f sinusidal signal. Th ral part f this functin is csω t whil th imaginary part f this functin is sinω t. pag 7 f 9 Cpyright Digilnt, Inc. ll rights rsrvd. Othr prduct and cmpany nams mntind may b tradmarks f thir rspctiv wnrs. j t ω

8 2.7.1: Sinusidal signals, cnmplx xpnntials, and phasrs Im sin t t R t tim t cs t tim Figur 5. Illustratin f j t ω. dditin and subtractin f cmplx numbrs is mst asily prfrmd in rctangular crdinats. Givn tw cmplx numbrs and, dfind as: a + c + jb jd th sum and diffrnc f th cmplx numbr can b dtrmind by: + ( a + c + j( b + d and ( a c + j( b d pag 8 f 9 Cpyright Digilnt, Inc. ll rights rsrvd. Othr prduct and cmpany nams mntind may b tradmarks f thir rspctiv wnrs.

9 2.7.1: Sinusidal signals, cnmplx xpnntials, and phasrs Multiplicatin and divisin, n th thr hand, ar prbably mst asily prfrmd using plar crdinats. If w dfin tw cmplx numbrs as: θ θ th prduct and diffrnc can b dtrmind by: j( θ + θ ( θ + θ and j( θ θ ( θ θ Th cnjugat f a cmplx numbr, dntd by a *, is btaind by changing th sign n th imaginary part f th numbr. Fr xampl, if a + jb, thn a jb Cnjugatin ds nt affct th magnitud f th cmplx numbr, but it changs th sign n th phas angl. It is asy t shw that 2 Svral usful rlatinships btwn plar and rctangular crdinat rprsntatins f cmplx numbrs ar prvidd blw. Th studnt is ncuragd t prv any that ar nt slf-vidnt. j 1 90 j j 1 90 j pag 9 f 9 Cpyright Digilnt, Inc. ll rights rsrvd. Othr prduct and cmpany nams mntind may b tradmarks f thir rspctiv wnrs.